Non - singular affine surfaces with self - maps

We classify surjective self-maps (of degree at least two) of affine surfaces according to the log Kodaira dimension. In this paper we are interested in the following question. Question. Classify all smooth affine surfaces X/C which admit a proper morphism f : X → X with degree f > 1. In [5] and [18], a classification of smooth projective surfaces with a self-map of degree > 1 has been given. This paper is inspired by their results. The case when X is singular appears to be quite hard so we restrict ourselves to the smooth case. Similarly, if f is not a proper morphism then again the problem is difficult. For example, we do not even know if there is anétale map of degree > 1 from C 2 to itself. This is the famous Jacobian Problem. If S is any Q-homology plane with κ(S) = −∞ then S admits an algebraic action of the additive group G a ([14]). Hence the automorphism group of such a surface is infinite. However, the problem of constructing a proper self-map of degree > 1 for S is quite non-trivial. Our main result can be stated as follows. Theorem. There is a complete classification of smooth complex affine surfaces X which admit a proper self-morphism of degree > 1, if either the logarithmic Kodaira dimension κ(X) ≥ 0 or the topological fundamental group π 1 (X) is infinite. 1 More precisely, any such X is isomorphic to a quotient of the form (∆ × A 1)/G or (∆ × C *)/G where ∆ is a smooth curve and G is a finite group acting freely on ∆ × A 1 or ∆ × C * respectively. As a consequence of the proof, we have: Corollary. Suppose that X is an affine surface with a proper morphism X → X of degree > 1. Then we have: (1) If κ(X) ≥ 0, then κ(X) = 0, 1 and X ∼ = (∆ × C *)/G where ∆ is a smooth affine curve and G is a finite group acting freely on ∆ × C *. (2) Suppose that κ(X) = −∞ and let ϕ : X → B be an A 1-fibration (for the existence see [15], Chapter I, §3). (2a) If κ(B) = −∞, then the topological fundamental group π 1 (X) is finite. (2b) If κ(B) = 0, then B ∼ = …